`BookProof.ChapterPaleyWienerSampling.BandLimited.hasSum_sinc` (hf : BandLimited T f) (x : ℝ) : HasSum (fun n : ℤ => f ((n : ℝ) / T) * (sinc (T * x - n) : ℝ)) (f x)

Proved

BookProof.ChapterPaleyWienerSampling.BandLimited.hasSum_sinc

by leonardopedro · Oct 20, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterPaleyWienerSampling.

BookProof.ChapterPaleyWienerSampling.BandLimited.hasSum_sinc (hf : BandLimited T f) (x : ℝ) : HasSum (fun n : ℤ => f ((n : ℝ) / T) * (sinc (T * x - n) : ℝ)) (f x)

Formalization note: Lean 4 identifier BookProof.ChapterPaleyWienerSampling.BandLimited.hasSum_sinc.

Preamble

-- Generated from ChapterPaleyWienerSampling.lean — theorem BookProof.ChapterPaleyWienerSampling.BandLimited.hasSum_sinc
import Mathlib
import Definitions.Def_ChapterPaleyWienerSampling
import Definitions.Def_ChapterShannonSampling
open BookProof.ChapterShannonSampling
open BookProof.ChapterPaleyWienerSampling
open BookProof.ChapterPaleyWienerSampling



open MeasureTheory Complex AddCircle Set
open scoped Real FourierTransform
open BookProof.ChapterShannonSampling (sinc bandSignal exists_rep half_add_period)

variable {T : ℝ} {f g : ℝ → ℂ}
variable {T : ℝ} [hT : Fact (0 < T)] {f g : ℝ → ℂ}

Formal statement

theorem BookProof.ChapterPaleyWienerSampling.BandLimited.hasSum_sinc (hf : BandLimited T f) (x : ℝ) :
    HasSum (fun n : ℤ => f ((n : ℝ) / T) * (sinc (T * x - n) : ℝ)) (f x) := by sorry

View graph